Team:Georgia State/Model

GSU iGEM

Predator-Prey Model

Predator-prey models can deal with the general loss-win interactions. Our group use predator-prey model to construct the relationship between two algaes, Oxyrrhis marina and Dunaliella tertiolecta.

The Lotka-Volterra Equations

The Lotka-Volterra model is a set of two first-order nonlinear differential equations that are used to represent the interactions between two species in which one is a predator and the other is the prey. These populations change with time according to the equations^[1]:

$\frac{dx}{dt}=\alpha x-\beta x y$

$\frac{dy}{dt}=\delta x y-\gamma y$

Where x represents the number of prey, y represents the number of predators, $\frac{dx}{dt}$ and $\frac{dy}{dt}$ represent the instantaneous growth rates for the two populations, and α, β, γ, δ are the parameters that describe the specific interactions between the predator and prey.

The Lotka-Volterra model also makes the following assumptions about the system’s environment and the evolution of the two species’ populations^[1]:

1. The prey will not starve
2. The predator population’s food supply completely depends on the prey population size
3. The rate of change of a population is proportional to its size
4. Predators have limitless appetite
5. The environment does not change in favor of one species

GSU iGEM decided that the Lotka-Volterra model was a perfect way to represent how our algae species interacted with one another.

Why a model?

The reason why a predator/prey model for Oxyrrhis marina and Dunaliella tertiolecta is being viewed in ASP-8A rather than F2 media (the media GSU iGEM decided to grow O. marina in) is because the team noticed than D. tertiolecta will begin to overgrow in ASP-8A media alongside the O. marina and eventually outcompete O. marina in the culture. This observation was made from June through August when the different types of media were being tested. The goal is to examine how much to feed the predator (O. marina) until the prey (D. tertiolecta) begins to overgrow.

Defining Our System

Figure.1 The basic model construction.

Parameters

D.T: Concentration of Dunaliella tertiolecta
O.M: Concentration of Oxyrrhis marina
a: Rate D.T is added to the system
b: D.T growth rate
c: Predation rate
d: O.M growth rate
f: O.M death rate

The model satisfies the identity:

Determining Our Rate Constants

To determine the constants we needed, we conducted an experiment

We start 5 different Dunaliella tertiolecta amount to feed in each 75ml flask. The start points are 0.25 ml, 0.5ml, 1.0ml, 1.5ml, 2.0ml of add-in D.T on concentration of 792,000/ml, and set all the O.M at the concentration of 1.2 x 10^4 cells/ml.

The two conditions start of 0.25ml and 0.5ml D.T are seem as the ideal condition that D.T will not overgrow and O.M concentration continues to increase.

Based on the net growth rate in 0.25ml and 0.5ml condition:

0.25 (net growth rate for DT)

9/10-9/13(3 days): 0

9/13-9/18(5 days): 4000/ml / 5days = 800/(day.ml)

9/18-9/20(2 days): (150000/ml - 4000/ml) / 2 days = 73000/(day.ml)

9/20-9/25(5 days): (1420000/ml - 150000/ml) / 5 days = 254000/(day.ml)

9/25-9/26(1 day): (890000/ml-1420000/ml) / 1 day = -530000/(day.ml) (delete)

9/26-10/4(8 days): (1180000/ml - 890000/ml) / 8 days = 36250/(day.ml)

Average growth rate = 91012.5 /(day.ml)

0.5 (net growth rate for DT)

9/10-9/13(3 days): (86000/ml - 42000/ml) / 3 days = 14667/(day.ml)

9/13-9/18(5 days): (332000/ml - 86000/ml) / 5days = 49200/(day.ml)

9/18-9/20(2 days): (150000/ml - 332000/ml) / 2 days =-91000/(day.ml) (delete)

9/20-9/25(5 days): (394000/ml - 332000/ml) / 5 days = 12400/(day.ml)

9/25-9/26(1 day): (658000/ml-394000/ml) / 1 day = 264000/(day.ml)

9/26-10/4(8 days): (3060000/ml - 658000/ml) / 8 days = 300250/(day.ml)

Average growth rate = 128103.4/(day.ml)

Final growth rate = (91012.5 /(day.ml) + 128103.4/(day.ml))/2 = 109557.95/(day.ml)

Thus, we get the D.T growth rate b is 109557.95/(day.ml).

We can rearrange the formulas as such:

Where g = d - f

From this we get the following:

For 1.0ml:

For 1.5ml:

For 2.0ml:

Thus, the predation rate c is 9.3/(day.ml), and the mixed O.M growth and death rate is -20890424/(day.ml).

Results

From the data we get above, we can simplify the model as the following:

An application of our model is to get the suitable initial concentration and feed in data for the D.T and O.M. Here is an example of the ideal initial and feed-in to reach a stable condition:

With an initial O.M concentration of 1.2 x 10^9 cells/ml and an initial D.T concentration of 19787 cells/ml and feeding with 2000 D.T cells/day, the environment can reach a stable condition.

[1] Beals, M., Gross, L., & Harrell, S. “Predator-Prey Dynamics: Lotka-Volterra.” National Institute for Mathematical and Biological Synthesis (1999).